U.S.-U.K. Cooperative Research: Applications of Brownian Motion to Differential Geometry

这个奖项为长期访问英国将 支持几何分析方面的合作研究， 波士顿的史蒂文·罗森伯格 University，K.D.沃里克大学的埃尔沃西， T.苏格兰爱丁堡大学的里昂说。 研究人员建议将概率和经典 微分几何问题的几何方法。 他们 将研究黎曼流形及其应用， 流形的拓扑和几何。 三个问题将是 研究：（1）Euler方程的Hopf猜想 负曲流形的特征;（2）一种方法， 推广了Bochner和Myers定理， 弯曲流形;（3）一种方法，用于产生软 拉普拉斯算子的特征值估计 福谷法。 后一个问题将随着 T医生爱丁堡大学的里昂。 歧管 这是许多数学领域的基本概念， 物理学 这种合作提供了一种创新的方法， 研究这些问题， 美国和英国调查人员在经典领域的专业知识 微分几何和概率论。

This award for a long term visit to the United Kingdom will support collaborative research in geometric analysis and differential geometry between Steven Rosenberg of Boston University, K.D. Elworthy of the University of Warwick, and T. Lyons of the University of Edinburgh, Scotland. The investigators propose to apply probabilistic and classical geometric methods to problems in differential geometry. They will study Riemannian manifolds and their applications to topology and geometry of manifolds. Three problems will be investigated: (1) The Hopf conjecture for the Euler characteristic of negatively curved manifolds; (2) a method for extending Bochner's and Myers' theorems for positively curved manifolds; and (3) a method for yeilding soft eigenvalue estimates for the Laplacian using a modification of the Fukaya method. The later problem will be developed with Dr. T. Lyons of the University of Edinburgh. Manifolds are a fundamental concept in many areas of mathematics and physics. This collaboration offers an innovative approach to studying these problems by combining the complementary expertise of the U.S. and British investigators in classical differential geometry and in probability respectively.